topic 3.1: kinematics - manitoba education and · pdf filedisplacement, velocity,...

TOPIC 3.1: KINEMATICSThe student will be able to:S3P-3-01: Differentiate between, and give examples of, scalar and vector

quantities. Examples: distance, speed, mass, time, temperature, volume, weight, position,displacement, velocity, acceleration, force…

S3P-3-02: Differentiate among position, displacement, and distance.S3P-3-03: Differentiate between the terms “an instant” and “an interval” of time. S3P-3-04: Analyze the relationships among position, velocity, acceleration, and

time for an object that is accelerating at a constant rate. Include: transformations of position-time, velocity-time, and acceleration-time graphsusing slopes and areas

S3P-3-05: Compare and contrast average and instantaneous velocity for non-uniform motion. Include: slopes of chords and tangents

S3P-3-06: Illustrate, using velocity-time graphs of uniformly accelerated motion,

that average velocity can be represented as and that

displacement can be calculated as

S3P-3-07: Solve problems, using combined forms of:

1 2 , , .2avg avg avg

v v d vv v at t

→ → → →→ → →+ ∆ ∆

= = =∆ ∆

1 2 .2

v vd t→ +

∆ = ∆

! !

avgdVt

→→ ∆

=∆

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Topic 3: Mechanics • SENIOR 3 PHYSICS

Topic 3.1 – 10

SPECIFIC LEARNING OUTCOME

S3P-3-03: Differentiate betweenthe terms “an instant” and “aninterval” of time.

GENERAL LEARNING OUTCOMECONNECTION

Students will…Understand how stability,motion, forces, and energytransfers and transformationsplay a role in a wide range ofnatural and constructedcontexts(GLO D4)

Entry-Level KnowledgeStudents may have some limited experiencefrom mathematics.

Notes to the Teacher The notion of time is important in physics.A common misconception is that an instantin time is a very short interval of time.However, an instant is considered to be asingle clock reading (t). If time were plottedon an axis, an instant is just a singlecoordinate along that axis. An interval is aduration in time (i.e., the intervalseparating two instants on the time axis[∆t]). The combination of a clock readingand instantaneous position is called anevent, a concept that later becomes usefulwhen introducing relativity.

Note: ∆, pronounced “delta,” is used torepresent the phrase “change in,” and iscalculated as “final – initial.”

e.g., ∆t is read as: the change in time(∆t = tfinal – tinitial)

∆v is read as: the change in velocity(∆v = vfinal – vinitial)

Class ActivityStudents time a runner at regular intervalsand make a data table of the time “splits”(i.e., the time at 10-m intervals). From thetable, note the instantaneous time and thetime intervals.

Senior Years Science Teachers’Handbook ActivitiesStudents use a Concept Frame for instantand interval of time.

SUGGESTIONS FOR INSTRUCTION

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Topic 3.1 – 11

SENIOR 3 PHYSICS • Topic 3: Mechanics

SKILLS AND ATTITUDES OUTCOME

S3P-0-2c: Formulate operationaldefinitions of major variablesor concepts.

Asking and Answering Questions Basedon DataStudents calculate ∆t using ∆t = tfinal – tinitialfrom the activity on the facing page.

Students differentiate, on a graph, betweenan instant and an interval of time.

SUGGESTIONS FOR INSTRUCTION SUGGESTIONS FOR ASSESSMENT

SUGGESTED LEARNING RESOURCES

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Topic 3.1 – 12


S3P-3-04: Analyze therelationships among position,velocity, acceleration, and timefor an object that is acceleratingat a constant rate. Include: transformations of position-time, velocity-time, and acceleration-time graphs using slopes and areas

SKILLS AND ATTITUDES OUTCOMES

S3P-0-2a: Select and useappropriate visual, numeric,graphical, and symbolicmodes of representation toidentify and representrelationships.

S3P-0-2d: Estimate and measureaccurately, using SystèmeInternational (SI) units.


Students will…Demonstrate appropriatescientific inquiry skills whenseeking answers to questions(GLO C2)

Entry-Level KnowledgeIn Senior 2 Science, students wereintroduced to uniform motion (S2-3-01) andaccelerated motion (S2-3-02 and S2-3-03).Senior 2 Science is intended to be a morequalitative treatment of motion developedwithin the context of the automobile. InSenior 3 Physics, the expectation is that amore sophisticated treatment of position,velocity, and acceleration will include amore complete graphical analysis withemphasis on the slope and arearelationships, and an introduction to themathematical relationships for motion.

Notes to the TeacherThe slope of a position-time graphrepresents velocity. The slope of a velocity-time graph represents acceleration.Alternatively, the area contained by aninterval of time in a velocity-time graphrepresents the displacement during the timeinterval. The area contained under aninterval of time in an acceleration-timegraph represents change in velocity over thetime interval. (See 3.14: Kinematics GraphsTransformation Organizer for a summarysheet.)

Begin this topic by interpreting the meaningof slope qualitatively, then by ratio. Given agraph of position versus time, an objectmoving faster will have a steeper slope. If

the graph is a straight line, then ∆d ∝ ∆tand ∆d = k∆t, where the constant k is theratio of displacement to the time interval(the slope). Examination of the ratio leads tothe conclusion that it is large when theobject is moving fast and small when it ismoving slow. The constant k represents theaverage velocity, and average velocity isdefined as the rate of change of positionwith respect to time. In formula notation:

Vavg =

The term “rate” refers to how much aquantity changes in one second. Thequantity for a linear mechanical system isdistance. Rate can be related to theequivalent for rotational (angle), fluid(volume or mass), electrical (charge), andthermal (heat) systems such that theconcept of rate becomes intuitive. Fluidrates might include the output of an oil wellin barrels per day. An example of electricalrate would be the amount of charge movingthrough the element of a toaster in C/s oramperes. Thermal rates, in joules persecond or BTUs per second, can be used todescribe the heating or cooling of our homes.

∆d∆t


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Topic 3.1 – 13


SKILLS AND ATTITUDES OUTCOMESS3P-0-2e: Evaluate the relevance,

reliability, and adequacy ofdata and data-collectionmethods.Include: discrepancies in dataand sources of error

S3P-0-2f: Record, organize, anddisplay data, using anappropriate format.Include: labelled diagrams,tables, graphs

S3P-0-2g: Interpret patterns andtrends in data, and infer orcalculate linear relationshipsamong variables.

S3P-0-4a: Demonstrate workhabits that ensure personalsafety, the safety of others,and consideration of theenvironment.



Acceleration can be investigated graphicallyby referring to a straight-line graph ofvelocity versus time. From this graph,∆d ∝ ∆t and ∆d = k∆t, where the constant kis the ratio of velocity change to the timeinterval. Examination of the ratio leads tothe conclusion that it is large when theobject’s velocity is changing rapidly andsmall when it is changing slowly. Theconstant k (slope) represents the averageacceleration of the object denoted by thesymbol a. We can now define averageacceleration as the rate of change of velocitywith respect to time. In formula notation:

aavg =

Galileo struggled with the definition ofacceleration as a rate of change of velocitywith respect to time (∆v/∆t) versus the rateof change of velocity with respect to position(∆v/∆d). Ultimately, Galileo chose ∆v/∆tbecause he was able to measure positionand time, and establish this as a powerrelationship.

∆v∆t

ObservationStudents take a pre/post test to examineknowledge of graphs and their relationshipsfor position, displacement, velocity, andacceleration.

Asking and Answering Questions Basedon DataGraphical Analysis: Students assess for anunderstanding of graphical representationsin the real world. Include graphicaldescriptions of motion.

Illustrative Example: Students describe themotion of an object, given a position-timegraph.

time (s)

posi

tion

(m)


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Topic 3.1 – 14








Prior Knowledge ActivitiesResearch shows students are easilyconfused by the concepts of position andvelocity, and velocity and acceleration. Byinvestigating velocity change with respect toposition as well as time, the student isforced to discriminate between the terms inthe same exercise. The idea of negativeacceleration is also often difficult to grasp.Students should work out all thepossibilities for negative acceleration whenthe velocity changes are positive and whenthe velocity changes are negative. Reviewall combinations.

After introducing the relationships amongposition, velocity, acceleration, and time,have students apply these formulas tosimple kinematics problems. Now is a goodtime for students to develop their skills forsolving algebraic equations. Present severaldifferent types of problems to enhance theseskills. See Appendix 3 for a complete list ofproblem types, solutions, and additionalsuggestions.

Laboratory ActivitiesSeveral types of activities may be used tostudy the relationships among position,velocity, acceleration, and time. Selectionwill depend on resources and the priorexperience of the students.

Students complete a table of values for anobject moving with a constant velocity of 5m/s, and then plot the points onto a positiontime graph (t = 0, 1, 2, 3, 4, 5). Repeat,using a velocity of 10 m/s, 2 m/s, and 0 m/s.Compare the slope of each line with thevelocity the slope represents.

Repeat the above exercise for velocity andtime, using various accelerations.

A student walks back and forth in a straightline, stopping now and then. Othermembers of the class record the position ofthe student at regular intervals of time.They gather this information, using metresticks and a stopwatch, or by using a motionsensor, then construct position-time,velocity-time, and acceleration-time graphs.

Other types of motion that could beinvestigated include walking away from thegroup at a steady pace, stoppingmomentarily, walking towards the group ata faster pace, rolling a ball down an incline,throwing a ball in the air, running 100 m, orany other suitable event.

Students can also investigate these kinds ofmotion by using microcomputers and motiondetectors, and analyzing the data by usinggraphical analysis software.


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Topic 3.1 – 15









A tickertape lab can be used to describe themotion of your hand or an object falling oraccelerating in some other manner.

Students examine the motion of an objectrolling across a table. Use a video camera torecord the motion with a scale (ruler)behind the object. Play the tape, stoppingand observing the equal displacements inequal time intervals. Fairly accurate datacan be collected by placing a transparencyonto the TV screen (static electricity willhold it in place) and plotting position with amarker. Many VCRs allow frame-by-frameviewing for this type of analysis. You mayalso want to calibrate your video camerawith a stopwatch to determine the intervalof time between successive frames.

DemonstrationsUse a rolling ball or cart with a numberscale in the background. Relate the scale tothe concepts of position, displacement, andintervals. A VCR with frame-by-frameadvance can be used to stop the motion.

Students can bring samples of children’stoys (either windup or electric) that willmove at a constant speed (or very nearlyconstant).

Research Report/PresentationStudents sketch a position-time graph of areal-world event (e.g., a runner, car, orrocket).

Lab Report Students examine tickertape motion forposition, velocity, and acceleration, takenfrom a lab activity.

Journal ReportStudents collect samples of different typesof motion from TV, movies, and cartoons.Students describe the motion and commenton the validity of the motion (e.g., thephysics of the Roadrunner cartoons).

Visual Displays Students describe the motion of a car frompoint A to point B, using position, velocity,and acceleration graphs.

Graphical Analysis Students make measurements on astroboscopic photograph, or sampletickertape, of a moving object.


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Topic 3.1 – 16








Throw a ball into the air, roll a ball down anincline, up an incline, across a horizontaland up an incline, et cetera. Sketch a graphof velocity-time and acceleration for eachcase.

Senior Years Science Teachers’Handbook ActivitiesStudents can complete a KWL/Knowledgechart (Senior Years Science Teachers’Handbook, page 9.13) to help assess theirunderstanding of ideas from Senior 2Science.

Journal EntryStudents construct a table of typical speedsfor familiar moving objects (e.g., a baseball,car, or snail). Sketch two graphs of your tripto school: a position-time and velocity-timegraph.

Students tell a story about motion (e.g.,describe the motion of a car acceleratingfrom rest at a stop sign to a constant speed).Include descriptions of the motion in thereal world, including sample data, a graphicpicture, and an algebraic representation.

Students make a list of objects that move ata constant speed (or very nearly constant).

Students identify other rates that they mayknow.

Students generate a list of objects thataccelerate at a constant rate.


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Topic 3.1 – 17










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JournalsHestenes, David, and Malcolm Wells. (1992)“A Mechanics Baseline Test.” The PhysicsTeacher 30.3: 159–166.

ResourcesAppendix 3.1: Working with the Modes ofRepresentationAppendix 3.5: Analysis of Data usingMicrosoft ExcelAppendix 3.6: Describing Motion in VariousWaysAppendix 3.7: Introducing Motion: Position,Time, Distance and Speed, Displacement,and VelocityAppendix 3.8: Motion: Interpreting Position-Time GraphsAppendix 3.9: Journal Entry: Kinematics(Position and Velocity)Appendix 3.10: Kinematics: Position,Velocity, and Acceleration GraphsAppendix 3.11: Kinematics and GraphingSkills BuilderAppendix 3.14: Kinematics GraphsTransformation Organizer

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Topic 3.1 – 18


S3P-3-05: Compare and contrastaverage and instantaneousvelocity for non-uniform motion. Include: slopes of chords andtangents



Entry-Level KnowledgeIn Senior 2 Science, students are introducedto these ideas qualitatively.

Notes to the TeacherA review of the graphical relationships forsteady and accelerated motion is a goodplace to begin. Emphasis on the slope of aline as a rate of change facilitates a greaterunderstanding of the meaning of velocityand acceleration. The meaning of an instantand an interval can lead into discussion tocompare average and instantaneousvelocity. The notion of a limit is difficult forstudents to understand. Consider the slopebetween two points on a curve as the points

become closer together, and relate this tothe average and instantaneous velocities. Asthe interval decreases, the line approachesthe tangent to the curve at that point.

Examine Diagram 5. To find averagevelocity, we need a time interval (i.e., twotimes). Plot a point on the curve at time 1and another point at time 2. Connect thetwo points and find the slope of the line.This will give you the average velocitybetween the two positions. To find thevelocity at a given time, plot a point on thecurve at that time. Then, draw a tangent tothe curve at that point (see Diagram 6). Theslope of this tangent is the instantaneousvelocity of the object at that given moment.


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Diagram 5 Diagram 6

Topic 3.1 – 19







Students will…Understand the composition ofthe Earth’s atmosphere,hydrosphere, and lithosphere,as well as the processesinvolved within and amongthem(GLO D5)

Class ActivityVideotape a ball falling under the influenceof gravity. Play back the tape in slowmotion or frame by frame and measure theposition of the ball for equal intervals oftime. The students can determine theaverage velocity of different time intervalsby determining the distance the ball fellbetween two consecutive frames andknowing the time between two consecutiveframes. The students can determine theinstantaneous velocity of the ball by plottinga distance-versus-time graph, connectingthe data points with a smooth curve, andfinding the slope of the tangents to thecurve at the desired times.

Students find velocity by graphical methods:a) Calculate average velocityb) Calculate instantaneous velocity



Appendix 3.8: Motion: Interpreting Position-Time GraphsAppendix 3.9: Journal Entry: Kinematics(Position and Velocity)

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Topic 3.1 – 20


S3P-3-06: Illustrate, usingvelocity-time graphs of uniformlyaccelerated motion, that averagevelocity can be represented as

and that displacement can becalculated as

1 2 .2

v vd t+∆ = ∆

! !!

avgdVt

∆=∆

!!



Notes to the TeacherFor a given time interval such that anobject begins with velocity v1 andaccelerates uniformly to velocity v2, thevelocity-time graph will be a straight-linesegment (Diagram 7). The midpoint of thissegment will be (v1 + v2)/2 (Diagram 8). Thearea under the trapezoid in Diagram 7 canbe shown to be the same as the area underthe rectangle created in Diagram 8.

To find the area of the trapezoid, one coulddraw a line at the midpoint between v1 andv2. This creates two triangles of equal area

Diagram 7

t1

v1

v2

t2

below and above the line. If we shift theupper triangle’s area into the lower triangle,a rectangle has been created (see Diagram8). Mathematically, the midpoint of the twovelocities is the average of v1 and v2 and:

(Recall that area under a velocity-timegraph results in displacement.)

Diagram 8

v1

v2

midpoint

t1 t2

1 2

1 2

area = height × base2

2

v vd t

v vd t

+ ∆ = = ∆

+ ∆ = ∆


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Topic 3.1 – 21



Students will…Employ effectivecommunication skills andutilize information technologyto gather and share scientificand technological ideas anddata(GLO C6)

Asking and Answering Questions Basedon DataGiven an equation, students sketch thecorresponding graph.



Appendix 3.10: Kinematics: Position,Velocity, and Acceleration Graphs

Appendix 3.11: Kinematics and GraphingSkills Builder

Appendix 3.14: Kinematics GraphsTransformation Organizer

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Topic 3.1 – 22


S3P-3-07: Solve problems, usingcombined forms of:

.avgvat

∆=∆

!"!

,avgdVt

∆=∆

"!""!1 2 ,

2avgv vV +

=

"! ""!""!



Notes to the TeacherInitially these formulas can be used to solvesingle-step problems to reinforce therepresentation for each variable, and tointroduce students to the concept of solvingproblems algebraically. These formulasshould then be applied to problems thatrequire two and three steps to solve.

Illustrative ExampleAn object moving at 3 m/s accelerates for 4 swith uniform acceleration of 2 m/s2. Findthe displacement moved.

Solution: State the given information insymbolic form:

v1 = +3 m/st = 4 sa = +2 m/s2

State the unknown:∆d = ?

Enough information is given to find v2using:

2 1

2 1

v v vat t

a t v v

∆ −= =∆ ∆

∆ = −

Solving for v2, we get:v2 = v1 + a∆tv2 = +3 + +2(4) = +11 m/s

Now, displacement can be found using:

Illustrative Example A car travelling at +10 m/s accelerates at arate of +3.0 m/s2 to a speed of +25 m/s.What distance does the car cover during theacceleration?

∆d =?v1 = +10 m/sv2 = +25 m/sa = +3.0 m/s2

1 2( )2

( 11 3) 42

28 m

v vd t

d

d

+∆ = ∆

+ + +∆ = ×

∆ = +


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Topic 3.1 – 23







Since any available version of adisplacement formula requires time, ∆t willhave to be found first:

Now that ∆t, has been found, we can find∆d as follows:

1 2

210 m/s 25 m/s 5.0 s

287.5 m 88 m

avgdvt

v vd t

d

d

∆=∆

+ ∆ = ∆ + + + ∆ = ×

∆ = + = +

2 1

225 m/s 10 m/s 5.0 s

3.0 m/s

vat

v vvta a

t

∆=∆

−∆∆ = =

+ − +∆ = =

Performance AssessmentStudent Assignments: It is recommendedthat students be given a significant amountof practice in the problem-solving area.Many are not used to this level ofmathematical treatment applied to physicalsituations.

Written Assessment While calculation problems may be used, itis important to assess in all modes ofrepresentation (i.e., the visual, numeric,graphical, symbolic, and linguistic).



See <www.physicsweb.com> for conceptdevelopment problems for kinematics,including graphical analysis and problemsolving.

Appendix 3.12: Kinematics: Position,Velocity, and Acceleration Graphs, and TheirEquations

Appendix 3.13: Kinematics Sampler: Graphs,Equations, and Problem Solving

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Topic 3.1 – 24


S3P-3-07: Solve problems, usingcombined forms of:

.avgvat

∆=∆

!"!

,avgdVt

∆=∆

"!""!1 2 ,

2avgv vV +

=

"! ""!""!



Illustrative ExampleIf an object is accelerated at +5.0 m/s2 andstarts from rest, what is its velocity after ithas travelled 20 m?

Solution: State the given information insymbolic form∆d = +20.0 m v1 = +0 m/sa = +5.0 m/s2 v2 = ?

No equation presented to date has enoughinformation to provide a direct numericalanswer in this scenario. Thus, twoequations will have to be solvedsimultaneously. Algebraically, there aremany ways to solve two equationssimultaneously. Students should bechallenged to reinforce the techniques theyacquire in math class. A comparison methodcan be used as follows:

(Since ∆t is not required for the finalanswer, set both equations equal to ∆t toeliminate the time variable.)

Equation 1:

1 2

1 2

22

v vd t

dtv v

+ ∆ = ∆ ∆

∆ = +

Equation 2:

Since both equations are equal to ∆t, theymust be equal to each other.

Senior Years Science Teachers’Handbook ActivitiesStudents write Process Notes to aid theiralgebraic solutions.

22

2 1

1 2

22

2

22

22 m2 s

2

2

02(20.0 m)0 5.0 m/s

40.0 m5.0 m/s200

14 m/s

v vdv v a

vv

vv

vv

−∆= +

−= +

=

=

= +

2 1

vatvt

av vt

a

∆=∆∆

∆ =

−∆ =


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Topic 3.1 – 25








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Topic 3.1 – 26��

NOTES

topic 3.1: kinematics - manitoba education and · pdf filedisplacement, velocity,...

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